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Solving the Problem of Dense Packing of Objects of Complex Geometry

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Advances in Mechanical Engineering

Abstract

The paper is devoted to the problem of dense packing of objects of arbitrary geometric shape with generalization in terms of dimension. A new solution method which consists in transforming the shape of objects by voxelization and the subsequent applying the developed algorithm for placing orthogonal polyhedrons is proposed. The effectiveness of the application of the developed method is demonstrated by an example of solving the problem of generating a layout of parts on the platform of a 3D printer. The application of the developed algorithms for solving the problem of dense packing of objects inside containers of arbitrary geometric shape is shown.

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References

  1. Leao, A.A., Toledo, F.M., Oliveira, J.F., Carravilla, M.A., Alvarez-Valdés, R.: Irregular packing problems: a review of mathematical models. Eur. J. Oper. Res. 282(3), 803–822 (2020). https://doi.org/10.1016/j.ejor.2019.04.045

    Article  MathSciNet  MATH  Google Scholar 

  2. Araújo, L.J., Özcan, E., Atkin, J.A., Baumers, M.: Analysis of irregular three-dimensional packing problems in additive manufacturing: a new taxonomy and dataset. Int. J. Prod. Res. 57(18), 5920–5934 (2019). https://doi.org/10.1080/00207543.2018.1534016

    Article  Google Scholar 

  3. Tahmasebi, P.: Packing of discrete and irregular particles. Comput. Geotech. 100, 52–61 (2018). https://doi.org/10.1016/j.compgeo.2018.03.011

    Article  Google Scholar 

  4. Stoyan, Y., Pankratov, A., Romanova, T.: Placement problems for irregular objects: mathematical modeling, optimization and applications. In: Butenko, S., Pardalos, P.M., Shylo, V. (eds.) Optimization Methods and Applications. SOIA, vol. 130, pp. 521–559. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68640-0_25

  5. Liu, X., Liu, J.M., Cao, A.X.: HAPE3D—a new constructive algorithm for the 3D irregular packing problem. Front. Inf. Technol. Electron. Eng. 16, 380–390 (2015). https://doi.org/10.1631/FITEE.1400421

    Article  Google Scholar 

  6. Chekanin, V.A., Chekanin, A.V.: Algorithm for the placement of orthogonal polyhedrons for the cutting and packing problems. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 41–48. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39500-1_5

    Chapter  Google Scholar 

  7. Chekanin, V.A., Chekanin, A.V.: Packing compaction algorithm for problems of resource placement optimization. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 1–9. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11981-2_1

    Chapter  Google Scholar 

  8. Johnson, D.S.: A brief history of NP-completeness, 1954–2012. Doc. Math. Extra Volume ISMP 359–376 (2012)

    Google Scholar 

  9. Verkhoturov, M., Petunin, A., Verkhoturova, G., Danilov, K., Kurennov, D.: The 3D object packing problem into a parallelepiped container based on discrete-logical representation. IFAC-PapersOnLine 49(12), 1–5 (2016). https://doi.org/10.1016/j.ifacol.2016.07.540

    Article  MathSciNet  Google Scholar 

  10. Kuprikov, M.Y., Markin, L.V.: Method of formalizing the layout of the internal compartments of aircraft. INCAS Bull. 11, 143–152 (2019). https://bulletin.incas.ro/files/kuprikov-m-y_markin__vol_11_special_issue_a_8.pdf

  11. Romanova, T., Bennell, J., Stoyan, Y., Pankratov, A.: Packing of concave polyhedra with continuous rotations using nonlinear optimization. Eur. J. Oper. Res. 268(1), 37–53 (2018). https://doi.org/10.1016/j.ejor.2018.01.025

    Article  MATH  Google Scholar 

  12. Romanova, T., Stoyan, Y., Pankratov, A., Litvinchev, I., Yanchevsky, I., Mozgova, I.: Optimal packing in additive manufacturing. IFAC-PapersOnLine 52(13), 2758–2763 (2019). https://doi.org/10.1016/j.ifacol.2019.11.625

    Article  Google Scholar 

  13. Stoyan, Y.G., Chugay, A.M.: Multistage approach to solving the optimization problem of packing nonconvex polyhedra. Cybern. Syst. Anal. 56(2), 259–268 (2020). https://doi.org/10.1007/s10559-020-00241-w

    Article  MATH  Google Scholar 

  14. Tolok, A.V., Tolok, N.B., Batuev, E.R.: Voxel modeling of the control of prototype manufacturing with additive technologies. Autom. Remote Control 82(3), 506–515 (2021). https://doi.org/10.1134/S0005117921030103

    Article  MATH  Google Scholar 

  15. Tolok, A.V., Tolok, N.B.: Mathematical programming problems solving by functional voxel method. Autom. Remote Control 79(9), 1703–1712 (2018). https://doi.org/10.1134/S0005117918090138

    Article  MathSciNet  Google Scholar 

  16. Verkhoturov, M., et al.: Optimization of placement in the tasks of rapid prototyping and manufacturing of volumetric parts based on additive technologies. In: ICCS-DE, pp. 298–305 (2020). CEUR-WS.org/Vol-2638/paper27.pdf

    Google Scholar 

  17. Byholm, T., Toivakka, M., Westerholm, J.: Effective packing of 3-dimensional voxel-based arbitrarily shaped particles. Powder Technol. 196(2), 139–146 (2009). https://doi.org/10.1016/j.powtec.2009.07.013

    Article  Google Scholar 

  18. Chekanin, V.A., Chekanin, A.V.: Solving the problem of decomposition of an orthogonal polyhedron of arbitrary dimension. In: Evgrafov, A.N. (ed.) MMESE 2020. LNME, pp. 52–59. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-62062-2_6

    Chapter  Google Scholar 

  19. Chekanin, V.A., Chekanin, A.V.: An efficient model for the orthogonal packing problem. In: Evgrafov, A. (ed.) Advances in Mechanical Engineering. LNME, pp. 33–38. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15684-2_5

    Chapter  Google Scholar 

  20. Chekanin, V.A., Chekanin, A.V.: New effective data structure for multidimensional optimization orthogonal packing problems. In: Evgrafov, A. (ed.) Advances in Mechanical Engineering. LNME, pp. 87–92. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29579-4_9

    Chapter  Google Scholar 

  21. Chekanin, V.A., Chekanin, A.V.: Deleting objects algorithm for the optimization of orthogonal packing problems. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 27–35. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53363-6_4

    Chapter  Google Scholar 

  22. Chekanin, V.A., Chekanin, A.V.: Algorithms for the formation of orthogonal polyhedrons of arbitrary dimension in the cutting and packing problems. Vest. MSTU «STANKIN» 3, 126–130 (2018). (in Russian)

    Google Scholar 

  23. Chekanin, V.A., Chekanin, A.V.: Algorithms for management objects in orthogonal packing problems. ARPN J. Eng. Appl. Sci. 11(13), 8436–8446 (2016). http://www.arpnjournals.org/jeas/research_papers/rp_2016/jeas_0716_4620.pdf

  24. Chekanin, V.A., Chekanin, A.V.: Design of library of metaheuristic algorithms for solving the problems of discrete optimization. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 25–32. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72929-9_4

    Chapter  Google Scholar 

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Chekanin, V.A., Chekanin, A.V. (2022). Solving the Problem of Dense Packing of Objects of Complex Geometry. In: Evgrafov, A.N. (eds) Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-91553-7_12

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  • DOI: https://doi.org/10.1007/978-3-030-91553-7_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-91552-0

  • Online ISBN: 978-3-030-91553-7

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